Logic of Ruler and Compass Constructions

Abstract

We describe a theory ECG of “Euclidean constructive geometry”. Things that ECG proves to exist can be constructed with ruler and compass. ECG permits us to make constructive distinctions between different forms of the parallel postulate. We show that Euclid’s version, which says that under certain circumstances two lines meet (i.e., a point of intersection exists) is not constructively equivalent to the more modern version, which makes no existence assertion but only says there cannot be two parallels to a given line. Non-constructivity in geometry corresponds to case distinctions requiring different constructions in each case; constructivity requires continuous dependence on parameters. We give continuous constructions where Euclid and Descartes did not supply them, culminating in geometrical definitions of addition and multiplication that do not depend on case distinctions. This enables us to reduce models of geometry to ordered field theory, as is usual in non-constructive geometry. The models of ECG include the set of pairs of Turing’s constructible real numbers [7].